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| In mathematics, a '''Minkowski plane''' (named after [[Hermann Minkowski]]) is one of the [[Benz plane]]s: [[Möbius plane]], [[Laguerre plane]] and Minkowski plane.
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| ==The classical real Minkowski plane==
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| [[Image:Minkowski-2d3d-model.png|400px|thumb|classical Minkowski plane: 2d/3d-model]]
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| Applying the [[pseudo-euclidean]] distance
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| <math>d_P(P_1,P_2)= (x_1-x_2)^2-(y_1-y_2)^2</math> on two points <math>P_i=(x_i.y_i)</math>
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| (instead of the euclidean one) we get the geometry of ''hyperbolas'', because
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| a pseudoeuclidean circle <math>\{P\in \R^2 \ | \ d_P(P,M)=r\}</math> is a
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| [[hyperbola]] with midpoint <math>M</math>. By a suitable coordinate transformation we can
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| rewrite the pseudo-euclidean distance as
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| <math>d'_P(P_1,P_2)=(x_1-x_2)(y_1-y_2)</math>. Now the hyperbolas have [[asymptotes]] parallel
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| to the coordinate axes. The following completion (see Moebius and
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| Laguerre planes) ''homogenizes'' the geometry of hyperbolas:
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| : <math>\mathcal P:=(\R\cup \{\infty\})^2=
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| \R^2 \cup (\{\infty\} \times\R) \cup (\R\times\{\infty\}) \
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| \cup \{(\infty,\infty)\} \ ,
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| \ \infty \notin \R</math>, the set of '''points''',
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| : <math>\mathcal Z:=\{\{(x,y)\in \R^2 \ | \ y=ax+b\}\cup\{(\infty,\infty)\} \ |
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| \ a,b \in \R, a\ne 0\}</math>
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| ::: <math>\cup \{\{(x,y)\in \R^2\ | y=\frac{a}{x-b}+c,x\ne b\}
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| \cup \{(b,\infty),(\infty,c)\} \ | \ a,b,c \in \R, a\ne 0\},</math> the set of '''cycles'''.
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| The [[incidence structure]] <math>({\mathcal P},{\mathcal Z},\in)</math> is called '''classical real Minkowski plane.'''
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| The set of points cosists of <math>\R^2</math> and two copies of <math>\R</math> and point <math>(\infty,\infty)</math>.<br />
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| Any line <math>y=ax+b ,a\ne0</math> is completed by point <math>(\infty,\infty)</math>, any hyperbola
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| <math> y=\frac{a}{x-b}+c,a\ne0 </math> by the two points <math>(b,\infty),(\infty,c)</math> (see figure).
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| Two points <math>(x_1,y_1)\ne(x_2,y_2)</math> can not be connected by a cycle if and only if
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| <math>x_1=x_2</math> or <math>y_1=y_2</math>. We define:<br />
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| Two points <math>P_1,P_2</math> are '''(+)-parallel''' (<math>P_1\parallel_+ P_2</math>) if <math>x_1=x_2</math> and '''(-)-parallel''' (<math>P_1\parallel_- P_2</math>) if <math>y_1=y_2</math>. <br />
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| Both these relations are [[equivalence relations]] on the set of points.<br />
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| Two points <math>P_1,P_2</math> are called '''parallel''' (<math>P_1\parallel P_2</math>) if
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| <math>P_1\parallel_+ P_2</math> or <math>P_1\parallel_- P_2</math>.
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| From the definition above we find:
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| '''Lemma:'''
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| :*For any pair of non parallel points <math>A,B</math> there is exactly one point <math>C</math> with <math>A\parallel_+ C \parallel_- B</math>.
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| :*For any point <math>P</math> and any cycle <math>z</math> there are exactly two points <math>A,B \in z</math> with <math>A\parallel_+ P \parallel_- B</math>.
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| :*For any three points <math>A,B,C</math>, pairwise non parallel, there is exactly one cycle <math>z</math> which contains <math>A,B,C</math>.
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| :*For any cycle <math>z</math>, any point <math>P\in z</math> and any point <math>Q, P \not\parallel Q</math> and <math>Q\notin z</math> there exists exactly one cycle <math>z'</math> such that <math>z\cap z'=\{P\}</math>, i.e. <math>z</math> '''touches''' <math>z'</math> at point P.
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| Like the classical Moebius and Laguerre planes Minkowski planes can be
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| described as the geometry of plane sections of a suitable quadric. But in this
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| case the quadric lives in '''projective''' 3-space: The classical real
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| Minkowski plane is isomorphic to the geometry of plane sections of a
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| [[hyperboloid of one sheet]] (not degenerated quadric of index 2).
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| ==The axioms of a Minkowski plane==
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| Let be <math>({\mathcal P},{\mathcal Z};\parallel_+,\parallel_-,\in)</math> an incidence structure with the set <math>\mathcal P</math> of points, the set
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| <math>\mathcal Z</math> of cycles and two equivalence relations <math>\parallel_+</math> ((+)-parallel) and <math>\parallel_-</math>
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| ((-)-parallel) on set <math>\mathcal P</math>.
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| For <math>P\in \mathcal P</math> we define:
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| <math>\overline{P}_+:=\{Q\in \mathcal P \ | \ Q\parallel_+ P\}</math> and
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| <math>\overline{P}_-:=\{Q\in \mathcal P \ | \ Q\parallel_- P\}</math>.
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| An equivalence class <math>\overline{P}_+</math> or <math>\overline{P}_-</math> is called '''(+)-generator'''
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| and '''(-)-generator''', respectively. (For the space model of the classical Minkowski plane a generator is a line on the hyperboloid.)<br />
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| Two points <math>A,B</math> are called '''parallel''' (<math>A\parallel B</math>) if <math>A\parallel_+ B</math> or <math>A \parallel_- B</math>.
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| An incidence structure <math>{\mathfrak M}:= ({\mathcal P},{\mathcal Z};\parallel_+,\parallel_-,\in)</math> is called '''Minkowski plane''' if the following axioms hold:<br />
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| [[File:Minkowski-axioms-c1-c2.png|thumb|Minkowski-axioms-c1-c2]]
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| [[File:Minkowski-axioms-c3-c4.png|thumb|Minkowski-axioms-c3-c4]]
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| : '''C1:''' For any pair of non parallel points <math>A,B</math> there is exactly one point <math>C</math> with <math>A\parallel_+ C \parallel_- B</math>.
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| :'''C2:''' For any point <math>P</math> and any cycle <math>z</math> there are exactly two points <math>A,B \in z</math> with <math>A\parallel_+ P \parallel_- B</math>.
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| :'''C3:''' For any three points <math>A,B,C</math>, pairwise non parallel, there is exactly one cycle <math>z</math> which contains <math>A,B,C</math>.
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| :'''C4:''' For any cycle <math>z</math>, any point <math>P\in z</math> and any point <math>Q, P \not\parallel Q</math> and <math>Q\notin z</math> there exists exactly one cycle <math>z'</math> such that <math>z\cap z'=\{P\}</math>, i.e. <math>z</math> '''touches''' <math>z'</math> at point P.
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| :'''C5:''' Any cycle contains at least 3 points. There is at least one cycle <math>z</math> and a point <math>P</math> not in <math>z</math>.
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| For investigations the following statements on parallel classes (equivalent to C1, C2 respectively) are advantageous.
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| :'''C1':''' For any two points <math>A,B</math> we have <math>|\overline{A}_+\cap\overline{B}_-|=1</math>.
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| :'''C2':''' For any point <math>P</math> and any cycle <math>z</math> we have: <math>|\overline{P}_+\cap z| = 1 = |\overline{P}_-\cap z|</math>.
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| First consequences of the axioms are
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| '''Lemma:''' For a Minkowski plane <math>{\mathfrak M}</math> the following is true
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| :a) Any point is contained in at least one cycle.
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| :b) Any generator contains at least 3 points.
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| :c) Two points can be connected by a cycle if and only if they are non parallel.
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| Analogously to Moebius and Laguerre planes we get the connection to the linear
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| geometry via the residues.
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| For a Minkowski plane <math>{\mathfrak M}=({\mathcal P},{\mathcal Z};\parallel_+,\parallel_-,\in)</math> and <math>P \in \mathcal P</math> we define the local structure
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| : <math>\mathfrak A_P:= (\mathcal P\setminus\overline{P},\{z\setminus\{\overline{P}\} \ | \ P\in z\in\mathcal Z\}
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| \cup \{E\setminus \overline{P} \ | \ E\in {\mathcal E}\setminus\{\overline{P}_+,\overline{P}_-\}\}, \in)</math>
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| and call it the '''residue at point P'''.
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| For the classical Minkowski plane <math>\mathfrak A_{(\infty,\infty)}</math> is the real affine plane <math>\R^2</math>.
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| An immediate consequence of axioms C1 - C4 and C1', C2' are the following two theorems.
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| '''Theorem:''' For a Minkowski plane <math>{\mathfrak M}=({\mathcal P},{\mathcal Z};\parallel_+,\parallel,\in)</math> any residue is an affine plane.
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| '''Theorem:'''
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| Let be <math>{\mathfrak M}=({\mathcal P},{\mathcal Z};\parallel_+,\parallel_-,\in)</math> an incidence structure with two equivalence relations <math>\parallel_+</math> and <math>\parallel_-</math> on the set <math>\mathcal P</math> of points (see above).
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| :<math>{\mathfrak M}</math> is a Minkowski plane if and only if for any point <math>P</math> the residue <math>\mathfrak A_P</math> is an affine plane.
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| The '''minimal model''' of a Minkowski plane can be established over the set
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| <math>\overline{K}:=\{0,1,\infty\}</math> of three elements:
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| : <math>\mathcal P:= \overline{K}^2,\qquad
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| \mathcal Z:= \{\{(a_1,b_1),(a_2,b_2),(a_3,b_3)\} \ | \ \{a_1,a_2,a_3\}=\{b_1,b_2,b_3\}=\overline{K}\}</math>,
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| : <math>(x_1,y_1)\parallel_+ (x_2,y_2)</math> if and only if <math>x_1=x_2 \ </math> and <math>\ (x_1,y_1)\parallel_- (x_2,y_2) \ </math> if and only if <math>\ y_1=y_2</math>.
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| Hence: <math>|\mathcal P|=9 </math> and <math>|\mathcal Z|=6</math>.
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| [[File:Minkowski-minimal-model.png|300px|thumb|Minkowski plane: minimal model]]
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| For finite Minkowski-planes we get from C1', C2':
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| '''Lemma:'''
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| Let be <math>{\mathfrak M}=({\mathcal P},{\mathcal Z};\parallel_+,\parallel_-,\in)</math> a finite Minkowski plane, i.e. <math>|\mathcal P| < \infty</math>. For any pair
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| of cycles <math>z_1,z_2</math> and any pair of generators <math>e_1,e_2</math> we have:
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| <math>|z_1|=|z_2|=|e_1|=|e_2|</math>.
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| This gives rise of the '''definition:'''<br />
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| For a finite Minkowski plane <math>{\mathfrak M}</math> and a cycle <math>z</math> of <math>{\mathfrak M}</math> we call the integer <math>n=|z|-1</math> the '''order''' of <math>{\mathfrak M}</math>.
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| Simple combinatorial considerations yield
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| '''Lemma:'''
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| For a finite Minkowski plane <math>{\mathfrak M}=({\mathcal P},{\mathcal Z};\parallel_+,\parallel_-,\in)</math> the following is true:
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| : a) Any residue (affine plane) has order <math>n</math>.
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| : b) <math>|\mathcal P|=(n+1)^2 \ </math>, c) <math>\ |\mathcal Z|=(n+1)n(n-1)</math>.
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| ==Miquelian Minkowski planes==
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| We get the most important examples of Minkowski planes by generalizing the
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| classical real model: Just replace <math>\R</math> by an arbitrary [[Field (mathematics)|field]]
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| <math>K</math> then we get '''in any case''' a Minkowski plane <math>{\mathfrak M}(K)=({\mathcal P},{\mathcal Z};\parallel_+,\parallel_-,\in)</math>.
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| Analogously to Moebius and Laguerre planes the Theorem of Miquel is a characteristic property of a Minkowski plane <math>\mathfrak M (K)</math> .
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| [[File:Theorem-of-miquel.png|300px|thumb|Theorem of Miquel]]
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| '''Theorem (MIQUEL):''' For the Minkowski plane <math>\mathfrak M (K)</math> the following is true:
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| : If for any 8 pairwise not parallel points <math>P_1,...,P_8 </math> which can be assigned to the vertices of a cube such that the points in 5 faces correspond to concyclical quadruples than the sixth quadruple of points is concyclical, too.
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| (For a better overview in the figure there are circles drawn instead of hyperbolas.)
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| '''Theorem (CHEN):''' Only a Minkowski plane <math>\mathfrak M (K)</math> satisfies the theorem of Miquel.
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| Because of the last Theorem <math>\mathfrak M(K) </math> is called a '''miquelian Minkowski plane'''.
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| '''Remark:''' The '''minimal model''' of a Minkowski plane is miquelian.
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| : It is isomorphic to the Minkowski plane <math>\mathfrak M(K) </math> with <math> K = GF(2)</math> (field <math>\{0,1\}</math>).
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| An astonishing result is
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| '''Theorem (Heise):''' Any Minkowski plane of ''even'' order is miquelian.
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| '''Remark:''' A suitable [[stereographic projection]] shows: <math>\mathfrak M(K) </math> is isomorphic
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| to the geometry of the plane sections on a hyperboloid of one sheet ([[quadric]] of index 2) in projective 3-space over field <math> K </math>.
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| '''Remark:''' There are a lot of Minkowski planes which are '''not miquelian''' (s. weblink below). But there are no "ovoidal Minkowski" planes, in difference to Möbius and Laguerre planes. Because any [[quadratic set]] of index 2 in projective 3-space is a quadric (see [[quadratic set]]).
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| ==References==
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| <references/>
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| *W. Benz, ''Vorlesungen über Geomerie der Algebren'', [[Springer Science+Business Media|Springer]] (1973)
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| *F. Buekenhout (ed.), ''Handbook of [[Incidence (geometry)|Incidence Geometry]]'', [[Elsevier]] (1995) ISBN 0-444-88355-X
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| ==External links==
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| * [http://eom.springer.de/b/b110290.htm Benz plane at SpringerLink]
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| *[http://www.mathematik.tu-darmstadt.de/~ehartmann/circlegeom.pdf Lecture Note '''''Planar Circle Geometries''''', an Introduction to Moebius-, Laguerre- and Minkowski Planes]
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| [[Category:Geometry]]
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